https://file.aiquickdraw.com/mathbot/file/1725409238283ltjdli7m.jpg

Let's analyze the functions and equations from the image. The function provided is:

\[ f(x) = 2x + 3 \]

You are asked to evaluate the function at specific inputs:

1. \( f\left(\frac{3}{2}\right) \)
2. \( f(a+h) \)
3. \( f(a) + f(h) \)

Let's solve each one:

1. **\( f\left(\frac{3}{2}\right) \):**
   \[
   f\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right) + 3 = 3 + 3 = 6
   \]
   
2. **\( f(a+h) \):**
   \[
   f(a+h) = 2(a+h) + 3 = 2a + 2h + 3
   \]

3. **\( f(a) + f(h) \):**
   \[
   f(a) = 2a + 3 \quad \text{and} \quad f(h) = 2h + 3
   \]
   \[
   f(a) + f(h) = (2a + 3) + (2h + 3) = 2a + 2h + 6
   \]

### Summary:
1. \( f\left(\frac{3}{2}\right) = 6 \)
2. \( f(a+h) = 2a + 2h + 3 \)
3. \( f(a) + f(h) = 2a + 2h + 6 \)

Would you like more details or have any further questions?

Here are some related questions to explore further:

1. What is the general formula for evaluating \( f(x) = mx + b \) at different values of \( x \)?
2. How does the result of \( f(a+h) \) compare with \( f(a) + f(h) \) for linear functions?
3. What would \( f(a-h) \) equal for the given function?
4. How would you find the inverse function of \( f(x) = 2x + 3 \)?
5. What would happen if the function were quadratic instead of linear?

**Tip:** For linear functions of the form \( f(x) = mx + b \), the function's behavior is entirely determined by the slope \( m \) and the intercept \( b \).

Evaluate the function f(x) = 2x + 3 for f(3/2), f(a+h), and f(a) + f(h).

Learn how to evaluate the linear function f(x) = 2x + 3 for different inputs like f(3/2), f(a+h), and f(a) + f(h). This problem covers key algebraic concepts suitable for students in Grades 7-10.

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